I was reading about wireless network cell planning for my home wifi (with $10-20 wifi routers, we're *all* wireless network engineers now!), and "distances" are sometimes "measured" in decibels: http://www.cisco.com/c/dam/en/us/td/i/400001-500000/400001-410000/400001-401... http://www.cisco.com/c/en/us/td/docs/wireless/controller/technotes/8-0/iPhon... Decibels are a logarithmic function; if x dB can reach distance D, then x+6 DB can reach distance 2*D. Assuming that wireless antennae emit radiation uniformly in all directions (very bad assumption!), then on a flat surface, one simple "cell" plan would place antennae in the centers of a uniform hexagonal grid. Of course, things are never this simple -- especially indoors. --- So the question is: are there interesting/elegant/surprising geometrical questions when the distances are measured with logarithms instead of normally? For example, some wireless questions involve having to "see" two cell towers simultaneously (e.g., wireless repeaters, roaming handoffs). In some cases, a roaming handoff may involve 2 signals which are exactly 6 dB apart. With normal distances, this locus would be a hyperbola, but since we're now talking about log distances, what is the locus then? --- Another problem involves distance/power/rate tradeoffs; this level of sophistication is beyond current home wifi, because typical home wifi systems can't use precise (and different) power levels & modulation schemes for each packet. https://en.wikipedia.org/wiki/Eb/N0 So as the distance increases, the data rate has to fall. So other loci involve these ratios; perhaps other interesting curves follow constant data rates, etc. --- Note that my interest here is *not* in the electrical engineering aspects of the system, but in the shapes of curves that might be generated by such calculations.